On the Total Curvature and Betti Numbers of Complex Projective Manifolds

Home , Betti number, Complex projective space, Total absolute curvature

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2018

On The Total Curvature And Betti Numbers Of Complex Projective Manifolds

Joseph Ansel Hoisington University of Pennsylvania, [email protected]

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This paper is posted at ScholarlyCommons. https://repository.upenn.edu/edissertations/2791 For more information, please contact [email protected]. On The Total Curvature And Betti Numbers Of Complex Projective Manifolds

Abstract We prove an inequality between the sum of the Betti numbers of a complex projective manifold and its total curvature, and we characterize the complex projective manifolds whose total curvature is minimal. These results extend the classical theorems of Chern and Lashof to complex projective space.

Degree Type Dissertation

Degree Name Doctor of Philosophy (PhD)

Graduate Group Mathematics

First Advisor Christopher B. Croke

Keywords Betti Number Estimates, Chern-Lashof Theorems, Complex Projective Manifolds, Total Curvature

Subject Categories Mathematics

This dissertation is available at ScholarlyCommons: https://repository.upenn.edu/edissertations/2791 ON THE TOTAL CURVATURE AND BETTI NUMBERS OF COMPLEX PROJECTIVE MANIFOLDS

Joseph Ansel Hoisington

A DISSERTATION

Mathematics

Presented to the Faculties of the University of Pennsylvania in Partial Fulﬁllment of the Requirements for the Degree of Doctor of Philosophy

2018

Supervisor of Dissertation

Christopher Croke, Professor of Mathematics

Graduate Group Chairperson

Wolfgang Ziller, Professor of Mathematics

Dissertation Committee: Herman Gluck, Professor of Mathematics Wolfgang Ziller, Professor of Mathematics Acknowledgments

I would like to express my sincere gratitude to Christopher Croke, my advisor, for his mentorship, encouragement and support, and for introducing me to the mathematics that led to these results. I would also like to thank Herman Gluck,

Peter McGrath, Tony Pantev, Brian Weber and Wolfgang Ziller for many helpful and enjoyable conversations.

ii ABSTRACT

ON THE TOTAL CURVATURE AND BETTI NUMBERS OF COMPLEX

PROJECTIVE MANIFOLDS

Joseph Ansel Hoisington

Christopher Croke

We prove an inequality between the sum of the Betti numbers of a complex projective manifold and its total curvature, and we characterize the complex projective manifolds whose total curvature is minimal. These results extend the classical theorems of Chern and Lashof to complex projective space.

iii Contents

1 Introduction 1

2 The Chern-Lashof Theorems and Their Spherical Formulations 9

3 Total Curvature and the Betti Numbers of Complex Projective

Manifolds 30

4 Complex Projective Manifolds with Minimal Total Curvature 51

5 Total Curvature and the Geometry of Complex Projective Hyper-

surfaces 58

iv Chapter 1

Introduction

The central results of this paper are an inequality between the total curvature of a complex projective manifold and its Betti numbers and a characterization of the complex projective manifolds whose total curvature is minimal. We will prove:

Theorem 1.1. Let M be a compact complex manifold, of complex dimension m, holomorphically immersed in complex projective space, and let T (M) be its total absolute curvature, as in Proposition 1.2 and Deﬁnition 1.4 below:

2m P m+1 A. Let βi be the Betti numbers of M with real coeﬃcients. Then βi ≤ ( 2 )T (M). i=0 In particular, T (M) ≥ 2.

B. If T (M) < 4, then in fact T (M) = 2. This occurs precisely if M is a linearly

embedded complex projective subspace.

The foundation for these results is a classical family of theorems that were

1 proved by Chern and Lashof. The definition of total absolute curvature for complex projective manifolds is based on an invariant which they defined, originally for submanifolds of Euclidean space, in [CL57] and [CL58]. We will define the total absolute curvature of a complex projective manifold in Definition 1.4 below. However, we will prove that the total absolute curvature of a complex projective manifold has the following geometric meaning:

Proposition 1.2. Let M be a compact complex manifold holomorphically immersed

π π < 2 in complex projective space. Let ν M be its normal disk bundle of radius 2 , and

⊥ < π N let Exp denote the normal exponential map from ν 2 M to CP . Let T (M) be its total absolute curvature. Then:

Z Z 2 ⊥ 2 ⊥ −1 < π N T (M) = |det(dExp )|dV ol 2 = ](Exp ) (q)dV ol P . V ol(CP N ) ν M V ol(CP N ) C

< π N ν 2 M CP

π In the ﬁrst expression, we integrate over the normal disk bundle of radius 2

π N ⊥ −1 because 2 is the diameter of CP . In the latter, ](Exp ) (q) denotes the pre- image count via Exp⊥ for a point q in CP N . Proposition 1.2 is equivalent to the meaning of Chern and Lashof’s invariant, which they deﬁned for a submanifold of

Euclidean space as follows:

Z 1 T (M) = det(A ) dV ol 1 . (1.1) V ol(SN−1) | ~u | ν M

ν1M

2 1 In this deﬁnition, ν M is the unit normal bundle of the immersion and A~u is the second fundamental form of the normal vector ~u. Dividing by V ol(SN−1) ensures that T (M) is the same whether we regard M as a submanifold of RN , or of a

0 higher-dimensional space RN+N containing RN as a linear subspace. Chern and

Lashof then proved:

Theorem 1.3 (Chern-Lashof Theorems, [CL57, CL58]). Let M be a closed manifold of dimension n immersed in Euclidean space:

A. (First Chern-Lashof Theorem) Let βi the Betti numbers of M, with coeﬃcients n P in the integers or any ﬁeld. Then βi ≤ T (M). In particular, T (M) ≥ 2. i=0

B. (Second Chern-Lashof Theorem) If T (M) < 3, then M is homeomorphic to

the n-sphere.

C. (Third Chern-Lashof Theorem) T (M) = 2 precisely if M is a convex hyper-

surface in an (n + 1)-dimensional linear subspace of RN .

In the introduction to their ﬁrst paper on total curvature, [CL57], Chern and

Lashof cite the theorems of Fenchel and F´ary-Milnor,in [Fe29] and [F´a49,Mi50], as motivation for their results. Fenchel’s theorem states that a smooth closed curve

γ in R3 has total curvature at least 2π, with equality precisely for plane convex curves. The Chern-Lashof theorems can be understood as a far-reaching general- ization of Fenchel’s theorem, to compact Euclidean submanifolds of any dimension and codimension. The F´ary-Milnortheorem states that if γ has total curvature at

3 most 4π, twice the minimum in Fenchel’s theorem, then it is unknotted. Part B of

Theorem 1.1 gives a similar statement for complex projective manifolds.

The deﬁnition of total absolute curvature for complex projective manifolds is an adaptation of Chern and Lashof’s invariant:

Deﬁnition 1.4 (Total Absolute Curvature of Complex Projective Manifolds). Let

M be a complex manifold, of complex dimension m, holomorphically immersed in the complex projective space CP N . Its total absolute curvature T (M) is deﬁned to be: Z (2N−2m−1) 2 sin r sin r π T (M) = |det( cos(r)IdTpM − A~v)| cos(r) dV < , V ol(CP N ) r r ν 2 M

< π ν 2 M π π < 2 where ν M is the normal disk bundle of radius 2 , A~v is the second fundamental

form of the normal vector ~v, r is its norm, and IdTpM : TpM → TpM is the identity transformation of the tangent space to M at its base point p.

Dividing by the volume of CP N in Deﬁnition 1.4 ensures that T (M) is the same whether we regard M as immersed in CP N , or in a higher-dimensional space

0 CP N+N containing CP N as a linear subvariety. The extra factor of 2 in Deﬁnition

1.4 will be explained in Proposition 3.1 and Remark 4.2. When we need to distinguish the invariant deﬁned for smooth submanifolds of Euclidean space in (1.1) from the invariant for complex projective manifolds in Deﬁnition 1.4, we will write

the ﬁrst as TRN (M) and the second as TCP N (M). In Deﬁnition 2.2, we will give a

4 formula for the total absolute curvature of smooth submanifolds of spheres, which we denote TSN (M).

Chern and Lashof’s invariant depends on the extrinsic geometry of a submanifold of Euclidean space. However, Calabi proved in [Ca53] that if a K¨ahlermanifold

(with a ﬁxed metric) admits a holomorphic isometric immersion into complex projective space, even locally, then any two holomorphic isometric immersions of this manifold into complex projective space are congruent by a holomorphic isometry of the ambient space. It follows that the total absolute curvature of a complex projective manifold is actually part of its intrinsic geometry. It would thus be interesting to ﬁnd a completely intrinsic representation of the total absolute curvature of a complex projective manifold. We will do so for curves in the complex projective plane:

Theorem 1.5. Let Σ be a smooth curve in CP 2, with K the sectional curvature of its projectively induced metric. Then: Z 1 (K − 4)2 + 4 T (Σ) = dA . π 6 − K Σ Σ

Note that if Σ is as above, then its sectional curvature is bounded above by 4, the holomorphic sectional curvature of the ambient space. This implies that the integrand in Theorem 1.5 is well-deﬁned. Theorem 1.5 is a corollary of a more

5 general result about the total absolute curvature of complex projective hypersurfaces which we will state and prove in Theorem 5.4. Theorem 1.5 implies several results about the total absolute curvature of smooth plane curves, which we state and prove in Proposition 5.8. In particular, the total absolute curvature of such a curve determines its degree:

Proposition 1.6. Let Σ be a smooth curve in CP 2. Then the degree of Σ is the unique d ∈ N such that 2d2 − 4d + 4 ≤ T (Σ) ≤ 2d2.

Theorem 1.5 also provides an example which shows that Part B of Theorem 1.1 is sharp, in Example 5.7.

We can summarize the proof of Theorem 1.1 as follows:

In Proposition 3.1, we will prove that if M is a complex manifold holomorphically immersed in the complex projective space CP N , and Mf is the S1 bundle over

M which is induced by the Hopf ﬁbration π : S2N+1 → CP N , and we immerse Mf

2N+1 N in S by lifting the immersion of M into CP , then TS2N+1 (Mf) = TCP N (M).

Chern and Lashof’s proof of Theorem 1.3 can be generalized to give similar theorems about submanifolds of any symmetric space - this was discovered by Koike in [Ko03] and [Ko05]. However, in almost all cases, these results are much narrower than those in Chern and Lashof’s original results. In particular, in complex projec-

6 tive space, these theorems only apply to submanifolds of real dimension 1 or less.

In spheres, on the other hand, Chern and Lashof’s proofs can be adapted to give results that are equivalent to their theorems in full generality. We will state and prove these results in Chapter 2. Because Chern and Lashof’s proofs work in such generality for submanifolds of spheres, and break down so completely in complex projective space, our strategy for proving the main theorems in this paper is to relate the geometry and topology of a complex projective manifold to those of its pre-image via the Hopf ﬁbration in the sphere.

We now give an outline of this paper:

In Chapter 2, we will review previous research related to total curvature, espe- cially in the complex projective setting. In addition to the Chern-Lashof theorems, this includes the results of Weyl on tube volumes in [We39], of Allendoerfer and

Fenchel on the higher-dimensional Gauss-Bonnet theorem in [Al40] and [Fe40], and more recent work of several authors. We will also state and prove a formulation of the Chern-Lashof theorems for submanifolds of spheres. In Chapters 3 and 4, we prove the main results in this paper: In Chapter 3, we will prove Part A of Theorem

1.1. This result follows from several other inequalities between the total curvature and Betti numbers of complex projective manifolds, which are generally stronger than Theorem 1.1.A - we will state and prove these results in Propositions 3.6 and

7 3.8. In Chapter 4, we prove Part B of Theorem 1.1 and discuss its relationship to several other results. We will also prove Proposition 1.2. In Chapter 5, we will study the total absolute curvature of complex projective hypersurfaces (complex projective manifolds of complex codimension 1) in greater detail. We will prove

Theorem 1.5 and discuss its implications. Throughout the paper, we will discuss possible directions for future research based on these results.

Throughout this paper, unless stated otherwise, RN and CP N will carry their canonical metrics, with the metric on CP N normalized to have holomorphic sectional curvature 4. Spheres will likewise carry their canonical metrics with constant curvature 1. Standard results and formulas from complex and algebraic geometry used in this paper can be found in [Huy06] and several other texts. Background about the local diﬀerential geometry of complex and K¨ahlersubmanifolds can be found in [Gr04].

8 Chapter 2

The Chern-Lashof Theorems and

Their Spherical Formulations

The proof of the Chern-Lashof theorems in [CL57] and [CL58] combines Morse theory, integral geometry, and a careful analysis of a Euclidean submanifold’s local extrinsic geometry. For any manifold M immersed in Euclidean space, of any dimension and codimension, one can deﬁne the Gauss map on the unit normal bundle

ν1M of the immersion. This map sends normal vectors ~u to their parallels in the unit sphere. A careful analysis of this map implies that for almost all ~w in the unit sphere, the height function h ~w in the direction ~w, when restricted to M, is a Morse function.

The upper bound for the sum of the Betti numbers in the ﬁrst Chern-Lashof

9 theorem is based on the fact that if f is a Morse function on a compact manifold M, counting its critical points gives an upper bound for the sum of the Betti numbers of M. The total absolute curvature of M is equal to the average number of critical points of the height functions h ~w, which are Morse functions for almost all ~w in the unit sphere, as described above. This average is greater than or equal to the sum of the Betti numbers of M. The second Chern-Lashof theorem is based on a theorem of Reeb, in [Re52]: that a compact manifold which supports a Morse function with only two critical points is homeomorphic to a sphere.

These observations are related to an extrinsic formulation of the Gauss-Bonnet-

Chern theorem, which was proved by Fenchel and Allendoerfer in [Fe40] and [Al40], building on earlier work of H. Hopf in [Ho26]. This formulation of the Gauss-

Bonnet-Chern theorem can also be proved from the same observations as the Chern-

Lashof theorems. In this proof, the Euler characteristic of a submanifold M of

Euclidean space arises as the sum of the critical points of the Morse functions h ~w, with each critical point signed by its index:

n n X i X i χ(M) = (−1) βi(M) = (−1) Ci, i=0 i=0

th where βi(M) is the i Betti number of M, with coeﬃcients in the integers or any ﬁeld, and Ci is the number of critical points of h ~w of index i. The corresponding fact in the Chern-Lashof theorems is that the sum of the Betti numbers of M can be bounded above by counting the critical points of a Morse function on M without

10 regard to their index:

n n X X βi(M) ≤ Ci. i=0 i=0

The extrinsic formulation of the Gauss-Bonnet-Chern theorem described above then says:

n Z X i 1 χ(M) = (−1) βi(M) = det(A~u)dV olν1M (~u). (2.1) V ol(SN−1) i=0

ν1M

The ﬁrst Chern-Lashof theorem says:

n Z X 1 βi(M) ≤ det(A~u) dV olν1M (~u). V ol(SN−1) | | i=0

ν1M

The result described above is genuinely the Gauss-Bonnet-Chern theorem in that for even-dimensional manifolds M, the integrand in (2.1) can be shown to coincide with the Pfaﬃan of the curvature forms, up to a normalization for the dimension of the ambient space. The proof of the Gauss-Bonnet-Chern theorem described above can be found in [Ba07]. The book [Wi82] by Willmore gives an overview of several branches of geometry in which the Chern-Lashof theorems have had a strong inﬂuence.

To our knowledge, the ﬁrst results about total absolute curvature in the complex projective setting are those of Ishihara in [Is86]. However, Milnor’s work in

[Mi63] and Thom’s work in [Th65] both use Morse theory to establish an inequality

11 between the degrees of real algebraic varieties and their Betti numbers. Based on their comments, both authors seem to have considered the possibility of extending their results using some of the observations we have used here, and Milnor uses some of these observations to extend his results to complex projective varieties. Cecil’s results in [Ce74] also involve the application of Morse theory to complex projective manifolds and use some of the same observations as the Chern-Lashof theorems.

Ishihara’s work gives a deﬁnition of total absolute curvature for a submanifold

M of complex projective space, and then relates this invariant to a family of maps from M to complex projective lines in the ambient space. In [AN95], Arnau and

Naveira define a family of total curvature invariants for submanifolds of complex projective space, which are adaptations of invariants defined for submanifolds of Eu- clidean space by Santalóin [Sa69, Sa70]. One of the invariants defined by Santaló coincides with the invariant defined by Chern and Lashof, and Arnau and Naveira show that Ishihara’s invariant coincides with one of their invariants. In [Ko03] and

[Ko05], Koike pursued the adaptation of the proof of the Chern-Lashof theorems to submanifolds of all symmetric spaces. However, in complex projective space these results are limited to submanifolds of real dimensions 0 and 1. This is an instance of a limitation that applies to Chern and Lashof’s proofs in all compact symmetric spaces except spheres - we will discuss this in Chapter 3. (The adaptations of

Chern and Lashof’s proofs to symmetric spaces of negative curvature are subject

12 to a diﬀerent set of limitations - see [Ko03].)

In addition to the Gauss-Bonnet-Chern theorem and these results about total

absolute curvature, there is another family of results that we believe provide im-

portant background for the results in this paper: In [We39], Hermann Weyl proved

that if M is a compact Riemannian manifold isometrically embedded in Euclidean

space, the volume of a small tubular neighborhood of M depends only on its intrinsic

geometry, not on the embedding. More precisely, he proved:

Theorem 2.1 (Weyl’s Tube Formula, [We39]). Let M be an n-dimensional compact

Riemannian manifold isometrically immersed in RN , and let ν

disk bundle of radius r. Let Exp⊥ be the normal exponential map from ν

RN . Then:

Z b n c 2 N−n 2 2i ⊥ ∗ (πr ) 2 X K2i(M)r (Exp ) (dV ol N ) = . R Γ( N−n+2 ) (N − n + 2)(N − n + 4) ··· (N − n + 2i) 2 i=0 ν

In the formula above, the coeﬃcients K2i(M) are calculated from the curvature

tensor of M.

If M is embedded in RN and r is chosen small enough that Exp⊥ : ν

is injective, then the formula above gives the volume of the tube of radius r about M.

This implies that, in this situation, the tube volume depends only on the intrinsic

13 geometry of M, not on the embedding.

Weyl’s proof of Theorem 2.1 generalizes immediately to submanifolds of spheres and hyperbolic spaces, and in [We39], he states and proves his result for submanifolds of spheres as well as Euclidean space. There is also a tube formula for complex submanifolds of complex projective space. To our knowledge, this was ﬁrst found independently by Wolf in [Wo71] and Flaherty in [Fl72]. Weyl’s tube formula was extended to compatible submanifolds of all rank-1 symmetric spaces, including complex submanifolds of CP N , by Gray and Vanhecke in [GV81]. The tube formula for complex projective manifolds can also be found in Alfred Gray’s book [Gr04].

Weyl’s tube formula, the Gauss-Bonnet-Chern theorem and the Chern-Lashof theorems are closely related. Total absolute curvature can be understood as a tube volume: For submanifolds of spheres and complex projective spaces, the total absolute curvature integrand coincides with the integrand in the tube formula for suﬃciently small normal vectors. In general, the total absolute curvature integrand is the absolute value of the integrand in the tube formula. The highest-degree term in Weyl’s tube formula is, up to a normalization, the Gauss-Bonnet-Chern integral, and the top-degree coeﬃcient in (2.2) is therefore given by the Euler characteristic of M - this fact played an essential part in Allendoerfer’s proof the Gauss-Bonnet-

Chern theorem in [Al40]. The history of this development can be found in [Gr04].

14 It is the close relationship between the Chern-Lashof theorems and Weyl’s tube formula, and the existence of a parallel to Weyl’s tube formula for complex projective manifolds in the results above, that motivated our search for an extension of the

Chern-Lashof theorems to complex projective space. We believe it would also be interesting to derive Gray and Vanhecke’s tube formulas for compatible submanifolds of complex and quaternionic projective space in [GV81] from Weyl’s tube formula for submanifolds of spheres in [We39] using Hopf ﬁbrations, as we have done here.

Because our results for complex projective manifolds depend on a formulation of the Chern-Lashof theorems for submanifolds of spheres, we will state and prove these results in the remainder of this chapter. The majority of these results are not new, but they are used in the proofs of our main theorems, and their proofs are helpful in many of our proofs in the rest of the paper. We will therefore give complete proofs of these results. The reader can also go ahead to the beginning of

Chapter 3 and return to these results as needed. The total absolute curvature of a submanifold of a sphere is deﬁned as follows:

Deﬁnition 2.2 (Total Absolute Curvature of Submanifolds of Spheres). Let M be an n-dimensional diﬀerentiable manifold immersed in the sphere SN . The total absolute curvature of T (M) of M is:

15 Z (N−n−1) 1 sin r sin r T (M) = det cos(r)Id − A dV ol <π . V ol(SN ) | ( TpM r ~v)| r ν M

ν<πM We integrate over ν<πM, the bundle of normal vectors of length less than π,

N because π is the diameter of S . A~v denotes the second fundamental form of the

normal vector ~v, r denotes its norm, and IdTpM : TpM → TpM is the identity transformation of the tangent space to M at its basepoint p. As in Deﬁnition 1.4,

N the normalization by V ol(S ) in Deﬁnition 2.2 ensures that TSN (M) = TSN+N0 (M) if SN is embedded as a totally geodesic submanifold of SN+N 0 .

For submanifolds of spheres, the diﬀerential of the normal exponential map can be expressed entirely in terms of the second fundamental form: Let ~u be a

N unit normal vector to a manifold M immersed in S and let A~u be its second fundamental form. Let e1, ..., en be a set of principal vectors for A~u with principal curvatures κ1, ..., κn, and let u2, ..., uN−n be an orthonormal basis for the subspace of νpM orthogonal to ~u. Then the diﬀerential of the normal exponential map at a normal vector ~v = r~u can be represented in this basis using the Jacobi ﬁelds of the sphere, as follows:

⊥ • dExp~v (ei) = (cos(r) − κi sin(r)) Ei(r), where Ei is the parallel vector ﬁeld

N with initial value ei along the geodesic γ~u of S through ~u.

⊥ sin r • dExp~v (uj) = r Fj, where Fj is the parallel vector ﬁeld along γ~u with

initial value uj.

16 ⊥ 0 • dExp~v (~u) = γ~u(r).

This gives us:

n ⊥ Q sin r (N−n−1) det(dExp )~v = (cos(r) − κi sin(r)) r i=1

sin r(N−n−1) = det cos(r)Id − sin(r)A . (2.3) TpM ~u r

Integrating |det(dExp⊥)| over a measurable subset of ν<πM deﬁnes a positive measure on ν<πM which is, in a natural sense, the pull-back of the measure on the ambient space SN via Exp⊥. Equation (2.3) implies that, up to the normalization by V ol(SN ), T (M) is the total mass of ν<πM with this measure. We record this result in the following proposition:

Proposition 2.3. Let M be a closed manifold immersed in the sphere SN . Then: Z 1 ⊥ T (M) = V ol(SN ) |det(dExp~v )|dV olν<π M (~v).

ν<πM

In Chern and Lashof’s original theorems, the equivalent statement is that for a submanifold M of RN , up to normalization by V ol(SN−1), T (M) is the total mass of the unit normal bundle ν1M, with the positive measure pulled back from the unit sphere SN−1 by the Gauss map.

It will be helpful to note that the total absolute curvature of a spherical sub-

17 manifold can also be written as follows: π

Z Z n 1 X i (N−n−1+i) (n−i) T (M) = (−1) sin (r) cos (r)σi(κ) dr dV olν1M (~u). V ol(SN ) | | i=0 ν1M 0 (2.4)

th Here, σi(κ) represents the i elementary symmetric function of the principal curvatures of the normal vector ~u. In particular, σ1(κ) is the mean curvature

κ1 + κ2 + ... + κn, σ2(κ) = κ1κ2 + κ1κ3 + ... + κn−1κn and σn(κ) = κ1κ2 . . . κn is the

Gauss curvature in the direction ~u, etc.

The spherical formulations of the Chern-Lashof theorems are as follows. These results can be found in the work of Koike:

Theorem 2.4 (Spherical Formulation of the Chern-Lashof Theorems - see [Ko05]).

Let M be an n-dimensional closed manifold isometrically immersed in the sphere

SN :

th A. Let βi be the i Betti number of M with coeﬃcients in the integers or any n P ﬁeld. Then βi ≤ T (M). In particular, T (M) ≥ 2. i=0

B. If T (M) < 3, then M is homeomorphic to Sn.

C. T (M) = 2 precisely if M is the boundary of a geodesic ball in an (n + 1)-

dimensional totally geodesic subsphere of SN .

18 The spherical Chern-Lashof theorems are based on the observation that if M is a closed manifold isometrically immersed in a round sphere SN , then for almost all q ∈ SN , the distance function from q is a Morse function when restricted on M.

The equivalent fact in the classical Chern-Lashof theorems is that almost all height

N−1 functions h ~w are Morse on M, for ~w in the unit sphere S in the ambient space

RN . As with submanifolds of Euclidean spaces, the total absolute curvature of a spherical submanifold is the average number of critical points of a family of Morse functions on M - in this case, of the distance functions dq. This average gives an upper bound for the sum of the Betti numbers of M, and if it is less than 3, it implies M is homeomorphic to a sphere by Reeb’s theorem.

We establish that almost all distance functions are Morse functions on M, and we derive a formula for their Hessians, in the next result:

Proposition 2.5. Let M be a closed manifold immersed in the sphere SN as in

Theorem 2.4. For almost all q ∈ SN , the distance function from q, when restricted to M, is a smooth Morse function.

Proof of Proposition 2.5. For the proof of this proposition and Lemma 2.6 below,

N we let deq denote dist(q, ·) as a function on S , and we let dq denote its restriction to M. For q not in the cut locus of any p ∈ M (i.e. for q 6∈ ±M), dq is a smooth function on M. That it is almost always a Morse function comes from the following:

⊥ <π Lemma 2.6. A point p of M is critical for dq iﬀ q = Exp (~v) for some ~v ∈ νp M.

19 ~v In that case, letting r = ||~v|| and ~u = r , Hess(dq) is diagonolized at p by a set of principal directions for the second fundamental form A~u, and the eigenvalue of

Hess(dq) corresponding to the principal curvature κ is cot(r) − κ.

Proof of Lemma 2.6. The gradient of dq on M is the orthogonal projection of the

N N gradient of deq on S . The gradient of deq at a pointq ¯ in S \{q, −q} is tangent to the minimizing geodesics from q toq ¯. grad(dq) is zero precisely where the minimizing geodesic from q is normal to M, so that q is in the image of the normal exponential map from p:

⊥ ˜ q = Exp ( − deq(p)grad(dq)).

Similarly, if Exp⊥(~v) = q, then the minimizing geodesic from q to p is normal to M at p, and p is critical for dq.

1 If p ∈ M and ~u ∈ νp M are as above, let e1, ..., en be a set of principal vectors

⊥ for A~u. For 0 < r < π, let q = Exp (r~u). We let grad(deq) and grad(dq) denote the

N > vector ﬁelds on S and M respectively, as above. Because grad(dq) = grad(deq) , along M we have:

⊥ grad(deq) = grad(dq) + grad(deq) .

A geodesic sphere of radius r in SN has principal curvature cot(r) in any tan-

SN gent direction, relative to the outward unit normal, so ∇e grad(deq) = cot(r)e for any e ∈ TpM, which will also be tangent to the geodesic sphere of radius r about

20 q = Exp⊥(r~u).

⊥ grad(dfq) Letting F be − ⊥ , F is a unit normal vector ﬁeld to M in a neighborhood ||grad(dfq) || of p, which coincides with ~u at p, so for each principal direction ei as above,

(∇SN F )> = −κ e . ei i i

We also have the following:

⊥ SN ei(||grad(dfq) ||) ⊥ 1 SN ⊥ ∇e F = ( ⊥ 2 )grad(deq) − ( ⊥ )∇e grad(deq) . i ||grad(dfq) || ||grad(dfq) || i

The tangential part of ∇SN F is therefore the tangential part of the expression ei above:

> > SN > 1 SN ⊥ 1 SN (∇e F ) = ( − ∇e (grad(deq) )) = ( − ∇e (grad(deq) − grad(dq))) i ⊥ i ⊥ i ||grad(dfq ) || ||grad(dfq ) ||

1 M 1 = ( ⊥ ) ∇e grad(dq) − cot(r)ei = ( ⊥ )(Hess(dq)(ei) − cot(r)ei) . ||grad(dfq) || i ||grad(dfq) ||

⊥ Noting that ||grad(deq) || = 1 at critical points of dq, we then have −κiei =

Hess(dq)(ei) − cot(r)ei, and therefore that Hess(dq)(ei) = (cot r − κi)ei. This completes the proof of Lemma 2.6.

The lemma implies that q = Exp⊥(r~u) has a degenerate critical point at p if and only if r = arccot κ for κ a principal curvature of M in the direction ~u. This is the same as the condition that the normal exponential map has a critical point

⊥ N at r~u, by the expression for det(dExp ) in (2.3). The q ∈ S for which dq is not

21 Morse are therefore the focal points of M in SN . These are the critical values of the normal exponential map, which are of measure zero in SN by Sard’s Theorem.

This completes the proof of Proposition 2.5.

Proof of Parts A and B of Theorem 2.4. By Proposition 2.5 and the lemma in its proof, the regular values of Exp⊥ : ν<πM → SN are precisely the points q ∈ SN for

N N which dq is a Morse function. Letting Sreg denote this subset of S , Sard’s Theorem

N N N implies that Sreg is of full measure in S . In fact, Sreg also contains an open, dense subset of SN .

⊥ This can be seen by extending Exp to be deﬁned on the bundle νMb over M, whose ﬁbre at p is the totally geodesic (N − n)-dimensional subsphere of SN or-

⊥ N thogonal to M at p. We denote this extension Expd : νMb → S . The critical ⊥ N points of Expd : νMb → S are closed in the compact manifold νMb , and thus a ⊥ N compact subset of νMb . Their image, the critical values of Expd : νMb → S , are a compact, and thus a closed subset of SN , whose complement is of full measure by

Sard’s Theorem. The critical values of Exp⊥ : ν<πM → SN are a subset of those

⊥ N ⊥ <π N of Expd : νMb → S . The regular values of Exp : ν M → S therefore contain ⊥ N N the regular values of Expd : νMb → S , which are open and dense in S .

As explained in the discussion before Proposition 2.3, integrating |det(dExp⊥)|

22 over neighborhoods of ν<πM deﬁnes a positive measure on ν<πM which is absolutely continuous with respect to dV olν<πM and is, in a natural sense, the pull-back of the measure on SN via Exp⊥. We will denote this measure by dµ. If φ is a measurable function on ν<πM, the integral of φ with respect to this measure is

⊥ given by integrating against |det(dExp )|dV olν<πM : Z Z ⊥ φ dµ = φ|det(dExp )|dV olν<πM .

ν<πM ν<πM

For any regular point ~v of Exp⊥ : ν<πM → SN , Exp⊥ is a local diﬀeomorphism

⊥ N from a neighborhood V of ~v to a neighborhood Q of Exp (~v) in Sreg. We then have: Z Z Z ⊥ ⊥ −1 φdµ = φ|det(dExp )|dV olν<πM = φ ◦ (Exp ) dV olSN . (2.5)

V V Q

N <π This implies that the pre-image of Sreg in ν M is of full measure relative to dµ: If ~v is a regular point of Exp⊥ whose image in SN is a critical value, then let V and Q be neighborhoods of ~v and Exp⊥(~v) with Exp⊥ : V → Q a diﬀeomorphism

N as in (2.5). Q ∩ Sreg contains a set which is open, dense and of full measure in Q, so

⊥ −1 N (Exp ) (Sreg) ∩ V likewise contains a set which is open, dense and of full measure in V . This implies that: Z ⊥ |det(dExp )|dV olν<πM = 0.

⊥ −1 N N (Exp ) (S \ Sreg) ∩ V

23 N N We can cover the regular points in the pre-image of S \ Sreg with open sets V as above. The integral of |det(dExp⊥)| over the critical points of Exp⊥ is zero, and

⊥ −1 N this implies that (Exp ) (Sreg) is of full measure for dµ.

We now apply the standard fact from Morse theory that if f is a Morse function

th on a closed manifold M, then βi(M; F ), the i Betti number of M with coeﬃcients in the ﬁeld F , is bounded above by Ci(f), the number of critical points of f which

N have index i. For q in Sreg, the distance function dq is Morse on M, and its critical points are in 1 − 1 correspondence with the pre-images of q via Exp⊥, as explained in Proposition 2.5 and Lemma 2.6. Letting Ci(dq) denote the number of critical

⊥ −1 points of dq of index i, and letting ](Exp ) (q) denote the pre-image count via

Exp⊥ for a point q of SN , we therefore have:

Z Z 1 ⊥ 1 ⊥ T (M) = det(dExp ) dV ol <π = det(dExp ) dV ol <π V ol(SN ) | ~v | ν M V ol(SN ) | ~v | ν M

<π ⊥ −1 N ν M (Exp ) (Sreg) Z Z n ! n 1 ⊥ −1 1 X X = ](Exp ) (q)dV ol N = C (d ) dV ol N ≥ β (M; F ) V ol(SN ) S V ol(SN ) i q S i i=0 i=0

N N Sreg Sreg

The ﬁrst equality above follows from Proposition 2.3 and the second from the

⊥ −1 N fact that (Exp ) (Sreg) is of full measure for dµ. The third follows from the change-of-variables formula (2.5) (with φ = 1), and the fourth from the correspondence between the pre-images of q via Exp⊥ and the critical points of the distance

24 N function dq on M, as in Lemma 2.6. We have also used the fact that Sreg is of full measure in SN .

n P This establishes Theorem 2.4.A, that βi(M; F ) ≤ T (M). i=0

If T (M) < 3, there must be a set S within SN of positive measure, which

N ⊥ −1 must therefore intersect Sreg, for which the pre-image count ](Exp ) (q) is less

N than 3. For any q0 ∈ S ∩ Sreg the pre-image count must therefore be equal to 2,

corresponding to the critical points of dq0 at its global minimum and maximum.

Reeb proved in [Re52] that if a closed manifold admits a Morse function with its global minimum and maximum as its only critical points, it is homeomorphic to a sphere (although it may not be diﬀeomorphic to the standard sphere.) For q0 as

above, dq0 provides such a Morse function, and this completes the proof of Theorem

2.4.B.

For odd-dimensional submanifolds of spheres, there is the following stronger version of Theorem 2.4.B. This will be important in proving Theorem 1.1.B:

Theorem 2.7. Let M 2m+1 be an odd-dimensional compact manifold, isometrically immersed in the sphere SN , with T (M) < 4. Then M is homeomorphic to the sphere S2m+1.

Proof. By the proofs of Theorem 2.4.A, Proposition 2.5 and Lemma 2.6, if T (M) <

25 2N+1 4, there is a point q0 in S such that the distance function dq0 is a Morse function

on M with fewer than 4 critical points. dq0 must have a critical point of index 0 at a point p1 of M where it realizes its global minimum, and a critical point of

index 2m + 1 at another point p2 where it attains its global maximum. If dq0 had a third critical point p3 of index k, it would have a fourth critical point p4 of index

2m + 1 − k because χ(M) = 0. This is impossible, so p1 and p2 are in fact the only

2m+1 critical points of dq0 on M, and M is homeomorphic to S by Reeb’s Theorem.

A similar statement holds for odd-dimensional manifolds immersed in Euclidean spaces.

We will prove Part C of Theorem 2.4 as a consequence of the following proposition. Parts A and B of Theorem 2.4 also follow from this proposition, however many of the observations in the proofs of Theorems 2.4.A and 2.4.B above will be important in explaining and proving our results for complex projective manifolds.

Proposition 2.8. For M isometrically immersed in SN and SN embedded as the unit sphere in RN+1,

TSN (M) = TRN+1 (M).

Proof. Let ν1M be the unit normal bundle of M in SN , and ν]1M its unit normal bundle in RN+1. Let ~ν be the outward unit normal vector to SN in RN+1. Any unit

26 normal vector ~z to M in RN+1 is of the form cos(θ)~ν + sin(θ)~u, for ~u a unit normal

N vector to M in S and θ ∈ [0, π]. Let Ag~u,θ be the second fundamental form of M

N+1 in R for a normal vector cos(θ)~ν + sin(θ)~u as above, and let A~u be the second fundamental form of M in SN for the corresponding ~u. For any principal vector e of A~u with principal curvature κ, we have the following identity :

> > N+1 N+1 N+1 R R R Ag~u,θ(e) = −(∇e ( cos(θ)~ν + sin(θ)~u)) = −( cos(θ)∇e ~ν + sin(θ)∇e ~u)

SN > = − cos(θ)e + sin(θ)(−∇e ~u) = (κ sin(θ) − cos(θ)) e.

e is therefore an eigenvector of Ag~u,θ with eigenvalue κ sin(θ) − cos(θ). Then we have: π

Z Z n 1 Y (N−n−1) TSN (M) = V ol(SN ) | (κi sin(θ) − cos(θ)) | sin (θ) dθ dV olν1M (~u) i=1

ν1M 0 Z Zπ 1 (N−n−1) = V ol(SN ) |det(Ag~u,θ)| sin (θ) dθ dV olν1M (~u)

ν1M 0 Z 1 = det(A ) dV ol (~z) = T N+1 (M). V ol(SN ) | f~z | ν]1M R

ν]1M

This proof has a simple geometric meaning:

27 <π N The image of νp M under the normal exponential map, as a subset of S , is

1 the same as the image of ν]p M under the Gauss map. In fact, these maps coincide

<π 1 under the natural identiﬁcation between ν M and ν]p M. Up to the normalization

N <π N by V ol(S ), TSN (M) is the mass of ν M with the measure pulled back from S by the normal exponential map, as in the proof of Proposition 2.3 and Theorems

2.4.A and 2.4.B. Similarly, TRN+1 (M) is the total mass of the unit normal bundle to M in RN+1, with the positive measure pulled back from SN by the Gauss map.

Because these maps coincide, the pulled-back measures coincide.

n N Proof of Part C of Theorem 2.4. Let M be a submanifold of S with TSN (M) =

2. By isometrically embedding SN in RN+1 as above, M is also a submanifold of

N+1 R with TRN+1 (M) = 2, and by Part C of Theorem 1.3, M is the boundary of a convex set in an aﬃne subspace A of RN+1 of dimension (n + 1). M is therefore a closed, embedded n-dimensional submanifold of A ∩ SN , which is homeomorphic to

Sn. M must therefore be equal to A ∩ SN . Let V be the unique (n + 2)-dimensional linear subspace of RN+1 containing the aﬃne subspace A. V ∩ SN is an (n + 1)- dimensional totally geodesic subsphere of SN , and M is embedded in V ∩ SN as the boundary of a geodesic ball.

To see the converse, if Sn is the boundary of a geodesic ball in a totally geodesic

(n + 1)-dimensional subsphere Sn+1 in SN , we isometrically embed Sn+1 as the unit sphere in Rn+2. This isometric embedding takes Sn to the boundary of a

28 geodesic ball about a point q in the unit sphere in Rn+2 - the boundary of such a geodesic ball is given by the intersection of the unit sphere Sn+1 with an (n + 1)- dimensional aﬃne subspace A of Rn+2. Therefore, by Proposition 2.8 and Theorem

n 1.3.C, TSn+1 (S ) = 2, and because the total absolute curvature of submanifolds of spheres is preserved under totally geodesic embeddings into higher-dimensional

n n spheres, TSN (S ) = TSn+1 (S ).

We end this chapter by noting that any compact Riemannian manifold can be isometrically embedded in a sphere of constant curvature 1 of sufficiently high dimension: The Nash embedding theorems, in [Na54] and [Na56], imply that every compact Riemannian manifold M can be isometrically embedded in a Euclidean space Rk of sufficiently high dimension. We can take such an embedding to have its image in a cell [0,D]k. Letting d be an integer greater than D2, the long diagonal of the unit cube in Rd has length greater than D, so [0,D]k, and M itself, can be isometrically embedded in the unit cube in Rdk. Finally, the unit cube in Rdk can be isometrically embedded in a fundamental domain for a flat torus in S2dk−1, so

M also admits such an embedding.

29 Chapter 3

Total Curvature and the Betti

Numbers of Complex Projective

Manifolds

We recall the deﬁnition of the total absolute curvature of a complex projective manifold as an integral in terms of its second fundamental form:

Deﬁnition 1.4 (Total Absolute Curvature of Complex Projective Manifolds)

Let M be a complex manifold, of complex dimension m, holomorphically immersed in the complex projective space CP N . Then its total absolute curvature is:

30 Z sin r sin r (2N−2m−1) 2 π T (M) = N |det cos(r)IdTpM − A~v | cos(r) dV ol < . V ol(CP ) r r ν 2 M

< π ν 2 M

It will be helpful to note that this can also be written as:

π Z Z 2 2 (2N−n−1) T (M) = N det cos(r)IdT M − sin(r)A~u cos(r) sin (r) dr dV ol 1 . V ol(CP ) | p | ν M

ν1M 0

One of the facts we will need to prove Theorem 1.1 is:

Proposition 3.1. Let M be a compact complex manifold holomorphically immersed in CP N , and let Mf be the circle bundle over M which is induced by the Hopf ﬁbration

π : S2N+1 → CP N . With the immersion of Mf in S2N+1 induced by the immersion of M in CP N , we have:

TS2N+1 (Mf) = TCP N (M).

Proposition 1.2 implies that the total absolute curvature of a complex projective manifold has the same meaning as the total absolute curvature deﬁned by Chern and Lashof in (1.1). However, the adaptation of Chern and Lashof’s proofs which we applied to submanifolds of spheres in Chapter 2 breaks down for submanifolds of complex projective space. This is because the cut locus of a point in complex projective space is a complex projective hyperplane, of real codimension 2.

31 If M is a submanifold of complex projective space which meets the cut locus of a point q ∈ CP N , the distance function from q may not be smooth on M - in particular, it may not be a Morse function. If M has real dimension 2 or greater, the union over the points of M of their cut loci in CP N has positive measure in

CP N . Because of this, total absolute curvature no longer gives an average for the number of critical points of a family of Morse functions on M. For compact complex submanifolds of CP N , we can be more precise about the scope of this limitation: such a manifold M necessarily meets the cut locus of every point q in CP N , because

N M intersects every linear subspace of CP of dimension N − dimC(M) or greater.

Because the Chern-Lashof theorems hold for submanifolds of spheres in full generality, as shown in Chapter 2, we will prove Theorem 1.1 by relating the geometry and topology of complex projective manifolds to those of their pre-images in spheres via the Hopf ﬁbration. Proposition 3.1 is the ﬁrst observation that we will need to do this - its proof is based on the following result, which gives a complete description of the second fundamental form of Mf in S2N+1.

Proposition 3.2. Let M be a complex manifold holomorphically immersed in CP N and Mf its pre-image in S2N+1 via the Hopf ﬁbration. Let ~u be a unit normal vector to M at p, e1, ..., en a set of principal vectors for the second fundamental form

A~u, and let κ1, ..., κn be the principal curvatures of e1, ..., en. Let u,e e1, ..., en be the horizontal lifts of ~u,e1, ..., en at any point pe in Mf above p.

32 Then ue is normal to Mf, e1, ..., en are principal vectors for the second fundamental form A , and each principal direction e has the same principal curvature κ as its ue ei i image ei. The tangent to the Hopf ﬁbre through pe is also a principal direction for

A with principal curvature 0. ue

2N+1 2N+2 N+1 Proof. We let νe denote the outward unit normal to S in R = C . We let J denote the complex structure of CP N and of M, and also of CN+1. We let

N hFS denote the canonical metric on CP , and h the induced metric on M, and we let eh denote the canonical metric on CN+1 and on the unit sphere S2N+1 in

N+1 2N+1 C . We let Ef0 denote the vector ﬁeld J(νe) on S . Ef0 is unit-length vector

field on S2N+1 tangent to the Hopf fibres. Its orthogonal complement in the tangent bundle of S2N+1 is invariant under the action of the complex structure of CN+1. The action of the complex structure of CN+1 on this subbundle of TS2N+1 induces the complex structure on CP N : Letting dπ denote the differential of the Hopf fibration

2N+1 N N π : S → CP , for any tangent vector ~v to CP , with ve any horizontal lift of

~v, we have:

J(~v) = dπ(J(ve))

For ei a principal vector for ~u with principal curvature κi as above, let Ei and

N U be unit-length vector ﬁelds on a neighborhood of CP which extend ei and ~u respectively, with Ei tangent and U normal to M. For any vector or vector ﬁeld

33 deﬁned on a neighborhood of CP N , let a tilde denote its horizontal lift to any neighborhood in S2N+1 via the Hopf ﬁbration.

By O’Neill’s formula,

2N+1 1 S ^CP N v ∇ Ue = ∇E U + [Eei, Ue] . (3.1) Eei i 2

N N N N Because ∇CP U = −A (e )+(∇CP U)⊥ = −κ e +(∇CP U)⊥, where (∇CP U)⊥ Ei ~u i Ei i i Ei Ei

N is the component of ∇CP U normal to M, and because the horizontal lift of Ei

N CP ⊥ ^CP N ⊥ ^CP N ⊥ −κiei + (∇ U) is given by −κiei + (∇ U) , where (∇ U) denotes the Ei e Ei Ei

N horizontal lift of (∇CP U)⊥, we have: Ei

2N+1 1 S ^CP N ⊥ v ∇ Ue = −κiei + (∇E U) + [Eei, Ue] . (3.2) Eei e i 2

^CP N ⊥ ^CP N ⊥ We note that (∇ U) is normal to M: For 1 ≤ l ≤ n, h((∇ U) , el) = Ei f e Ei e

N h ((∇CP U)⊥, e ) = 0. We also have h((∇^CP N U)⊥, E ) = 0 because (∇^CP N U)⊥ is FS Ei l e Ei e0 Ei

2N+1 N horizontal and Ee0 is vertical for the submersion π : S → CP .

Noting that vertical directions for the Hopf ﬁbration π : S2N+1 → CP N are tangent to Mf, we can rewrite (3.1) and (3.2) as follows:

S2N+1 1 v S2N+1 ⊥ ∇ Ue = −κiei + ( )[Eei, Ue] + (∇ Ue) . Eei e 2 Eei

2N+1 ⊥ 2N+1 (∇S Ue) denotes the component of ∇S Ue normal to Mf, which is equal to Eei Eei

2N+1 2N+1 ^CP N ⊥ S S ⊥ (∇E U) . Noting also that ∇ Ue = −Au(ei) + (∇ Ue) , where Au is the i Eei e e Eei e

34 second fundamental form of Mf in the normal direction ue, we infer that:

A (e ) = κ e − 1 [E , U]v. ue ei iei 2 ei e

The proof that ei is a principal vector for ue, with principal curvature κi, will be

v complete once we show that [Eei, Ue] is zero.

v Because the vertical distribution for the Hopf ﬁbration is spanned by Ee0,[Eei, Ue]

2N+1 is equal to eh([Eei, Ue], Ee0)Ee0. We write this in terms of the connection in S as follows:

S2N+1 S2N+1 eh([Eei, Ue], Ee0)Ee0 = eh(∇ Ue − ∇ Eei, Ee0)Ee0 Eei Ue

S2N+1 S2N+1 = (eh(∇ U,e Ee0) − eh(, ∇ Eei, Ee0))Ee0. (3.3) Eei Ue

N+1 Because the Euclidean metric on C is Hermitian and Ee0 = J(νe), we have:

2N+1 N+1 N+1 N+1 S C C C eh(∇ U,e Ee0) = eh(∇ U,e Ee0) = eh(J(∇ Ue),J(Ee0)) = −eh(J(∇ Ue), ν). Eei Eei Eei Eei e

N+1 Because the metric on CN+1 is K¨ahler,its connection ∇C commutes with the complex structure J, so letting J(Ue) denote the vector ﬁeld which results from applying the complex structure of CN+1 to Ue, we have:

N+1 N+1 N+1 C C C eh(J(∇ Ue), ν) = eh(∇ J(Ue), ν) = Eei(eh(J(Ue), ν)) − eh(J(Ue), ∇ ν). Eei e Eei e e Eei e

N+1 2N+1 C Because ν is the outward unit normal to S , ∇ ν = Eei, so the above is e Eei e equal to:

Eei(eh(J(Ue), νe)) − eh(J(Ue), Eei).

35 Because M is a complex submanifold of CP N , its tangent and normal spaces are preserved by the complex structure of CP N . Letting J(U) denote the vector

ﬁeld which results from applying the complex structure of CP N to U, we therefore have that J(U) is normal to M. Because the complex structure on CN+1 preserves

2N+1 the subbundle of TS orthogonal to Ee0, J(Ue) is a horizontal vector ﬁeld for the Hopf ﬁbration. And because the complex structure on CP N is induced by the action of the complex structure of CN+1 as described above, J(Ue) is the horizontal lift of J(U).

Applying this to the formula above, we have that eh(J(Ue), Eei) = hFS(J(U),Ei) =

N 0 because J(U) is normal to M and Ei is tangent to M in CP . We have

2N+1 eh(J(Ue), νe) ≡ 0 because J(Ue) is tangent and νe normal to S . This then implies

S2N+1 that Eei(eh(J(Ue), ν)) is zero. And this implies that the expression eh(∇ U,e Ee0) e Eei in (3.3) is zero.

S2N+1 We can apply the same observations to show that the term eh(∇ Ei, E0) in Ue e e (3.3) is also zero:

2N+1 N+1 N+1 S C C eh(∇ Ei, E0) = eh(∇ Ei, E0) = eh(J(∇ Ei),J(E0)). Ue e e Ue e e Ue e e

This is because the Euclidean metric is Hermitian. Applying the identity Ee0 =

J(νe), the fact that the connection and complex structure of the Euclidean metric

N+1 commute and the fact that ∇C ν = U, we have that this is equal to: Ue e e

36 N+1 N+1 −eh(J(∇C Eei), ν) = eh(J(Eei), ∇C ν) − Ue(eh(J(Eei), ν)) = eh(J(Eei), Ue) − Ue(eh(J(Eei), ν)) = 0. Ue e Ue e e e

This completes the proof that e is a principal vector for A , with principal cur- ei ue vature κi, just as ei is a principal vector for A~u with prinicipal curvature κi.

To see that the tangent to the Hopf ﬁbre is also a principal vector, with principal curvature zero, we note that:

S2N+1 S2N+1 eh(∇ U,e Ee0) = Ef0(eh(U,e Ef0)) − eh(U,e ∇ Ee0). Ee0 Ef0

2N+1 S2N+1 Because the Hopf ﬁbres are geodesics of S , we have ∇ Ee0 = 0. We also Ee0 have eh(U,e Ee0) ≡ 0 because Ue is normal and Ee0 tangent to Mf. This implies that

S2N+1 Ef0(eh(U,e Ef0)) = 0, and as a consequence, eh(∇ U,e Ee0) = 0. Ee0

Letting Eei be a horizontal lift of a vector ﬁeld Ei as above for i = 1, 2, ··· , n, we

S2N+1 S2N+1 have that eh(∇ U,e Eei) = Ee0(eh(U,e Eei)) − eh(U,e ∇ Efi). Ee0(eh(U,e Eei)) is zero Ee0 Ef0 because eh(U,e Eei) is zero, so we are left with:

S2N+1 S2N+1 −eh(U,e ∇ Efi) = eh(U,e [Eei, Ee0] − ∇ Ef0). Ef0 Efi

eh(U,e [Eei, Ee0]) is zero because Ue is normal and [Eei, Ee0] is tangent to Mf, so this

S2N+1 leaves us with eh(U,e ∇ Ef0). This is equal to eh(Au(ei), Ee0). Because ei is a princi- Efi e e e pal vector for A , as we showed above, this is equal to h(κ e , E ) = κ h(e , E ) = 0. ue e iei e0 ie ei e0

S2N+1 This establishes that eh(∇ U,e Eei) = 0 and completes the proof that the Hopf Ee0

ﬁbres are principal directions for Mf in S2N+1, with principal curvature zero.

37 The proof of Proposition 3.1 also uses the following result, which gives a simpler

expression for TCP N (M) and is important in some of our later results.

Proposition 3.3. Let M be a complex submanifold of a K¨ahlermanifold P . Sup- pose ~u is a unit normal vector to M at p and e is an principal vector for the second fundamental form A~u, with principal curvature κ. Let J denote the complex structure. Then J(e) is also a principal vector for A~u, with principal curvature −κ.

Proof. Let h denote the metric on P and the induced metric on M, and let e1, ..., en be an orthonormal basis for TpM with e1 = e and e2 = J(e). We are trying to show

P that h(∇e2 ~u,ej) = κ if j = 2, and is zero otherwise.

Writing the symmetric bilinear form associated to A~u as B~u, we have:

h(∇P ~u,e ) = −B (e , e ) = −B (e , e ) = h(∇P ~u,e ). e2 j ~u 2 j ~u j 2 ej 2

Because the metric is Hermitian and e2 = J(e1), we have:

h(∇P ~u,e ) = h(∇P ~u,J(e )) = −h(J(∇P ~u), e ). ej 2 ej 1 ej 1

Because the metric is K¨ahler,we have:

−h(J(∇P ~u), e ) = −h(∇P J(~u), e ). ej 1 ej 1

38 J(~u) is a normal vector to M because M is a complex submanifold of P . Written in terms of the bilinear second fundamental form, the expression above is then equal to:

P BJ(~u)(ej, e1) = BJ(~u)(e1, ej) = −h(∇e1 J(~u), ej).

Using again the fact that the metric is K¨ahler,we have:

P P −h(∇e1 J(~u), ej) = −h(J(∇e1 ~u), ej).

Because e1 is a principal vector for ~u with principal curvature κ, we have:

P −h(J(∇e1 ~u), ej) = h(J(κe1), ej) = κh(J(e1), ej) = κh(e2, ej).

This will be κ if j = 2, and zero otherwise.

For a complex projective manifold M of complex dimension m, we can thus write its principal curvatures for a normal vector ~u as κ1, −κ1, κ2, −κ2, ··· , κm, −κm. We then have:

m Y 2 2 2 det( cos(r)IdTpM − sin(r)A~u) = ( cos (r) − κi sin (r)). (3.4) i=1

And we have the following two formulae for TCP N (M): π Z Z2 m 2 Y 2 2 2 (2N−2m−1) T P N (M) = | ( cos (r) − κi sin (r))| cos(r) sin (r)dr dV olν1M C V ol(CP N ) i=1

ν1M 0 (3.5)

39 π Z Z2 m 2 X i (2N−2m−1+2i) (2m−2i+1) 2 T P N (M) = | (−1) sin (r) cos (r)σi(κ )|dr dV olν1M . C V ol(CP N ) i=0

ν1M 0 (3.6)

2 th Here, as in (2.4), σi(κ ) represents the i elementary symmetric function of the

2 squares of the principal curvatures of the normal vector ~u. In this case, σm(κ ) =

2 2 2 m κ1κ2 . . . κm is (−1) times the Gauss curvature.

Remark 3.4. Proposition 3.3 implies that K¨ahlersubmanifolds are minimal. In fact, closed K¨ahlersubmanifolds are minimal in the very strong sense that they have the minimal volume of any submanifold in their homology class. This is known as

Wirtinger’s inequality (not to be confused with the more famous inequality for periodic functions) and a proof can be found in [Fr65].

2N+1 Proof of Proposition 3.1. Let ue be a unit normal vector to Mf in S , with ~u its image via the differential of the Hopf fibration. ~u is normal to M in CP N . We let κe0, κe1, κe2, ··· κen be the principal curvatures of ue, with κe0 = 0 corresponding to the principal direction along the Hopf fibre. We let κ1, −κ1, κ2, −κ2, ··· , κm, −κm denote the principal curvatures of ~u, so that we have:

κe0 = 0, κe1 = κ1, κe2 = −κ1, κe3 = κ2, κe4 = −κ2, ··· , κen−1 = κm, κen = −κm.

Let pe denote the basepoint of ue in Mf and p its image in M. For r ∈ [0, π], we have:

40 n Y det( cos(r)IdT M − sin(r)Au) = ( cos(r) − κj sin(r)) pe f e e j=0

m Y 2 2 2 = cos(r) ( cos (r) − κi sin (r)) = cos(r)det( cos(r)IdTpM − sin(r)A~u). (3.7) i=1

π Z 2 (2N−n−1) = 2 |det( cos(r)IdT M − sin(r)Au)| sin (r)dr. pe f e

Equation (3.7) implies this is equal to the corresponding integral for ~u:

π Z 2 (2N−n−1) 2 |det( cos(r)IdT M − sin(r)Au)| sin (r)dr pe f e

π Z 2 (2N−n−1) = 2 |det( cos(r)IdTpM − sin(r)A~u)| cos(r) sin (r)dr.

(The additional term cos(r) in the integrand for TCP N (M), which appears in the second term above, is because of the Jacobi ﬁeld with initial value J(~u) along the

41 N geodesic γ~u in CP . This is explained in the proof of Proposition 1.2, in Chapter 4.)

The Hopf ﬁbres are geodesics of S2N+1 of length 2π. For p ∈ M, we therefore have: π Z Z2 (2N−n−1) 2π |det( cos(r)IdTpM − sin(r)A~u)| cos(r) sin (r)dr d~u

1 νpM 0 π Z Z Z2 (2N−n−1) = |det( cos(r)IdT M − sin(r)Au)| sin (r)dr du dp. pe f e e e

−1 1 π (p) ν Mf pe 0

Together with the fact that V ol(S2N+1) = 2πV ol(CP N ), this implies that:

π Z Z2 2 (2N−n−1) T P N (M) = N det( cos(r)IdT M − sin(r)A~u) cos(r) sin (r)dr dV olν1M . C V ol(CP ) | p |

ν1M 0 π Z Z 1 (2N−n−1) = V ol(S2N+1) |det( cos(r)IdT M − sin(r)Au)| sin (r)dr dV olν1M = TS2N+1 (Mf). pe f e f

1 ν Mf 0

In light of Proposition 3.1, we can establish an inequality between the total absolute curvature of a complex projective manifold and its Betti numbers with the following result:

42 Proposition 3.5. Let M be a compact complex manifold, holomorphically immersed in the complex projective space CP N . Let Mf be the S1-bundle over M which is induced by the immersion of M into CP N from the Hopf ﬁbration S2N+1 → CP N .

For 0 ≤ k ≤ m = dimC(M), we have: b k c P2 βk(M; R) = βk(Mf; R) + βk−2(Mf; R) + βk−4(Mf; R) + ··· = βk−2i(Mf; R). i=0

For m ≤ k ≤ n = dimR(M), we have: b n−k c P2 βk(M; R) = βk+1(Mf; R)+βk+3(Mf; R)+βk+5(Mf; R)+··· = βk+2i+1(Mf; R). i=0

1 Proof. Let πe : Mf → M denote the S -bundle as above. The ﬁbres of Mf are oriented by the 1-form dθ dual to the Hopf vector ﬁeld, and Mf is oriented by

∗ dθ ∧ πe (dV olM ). The cohomology of Mf is therefore related to the cohomology of

M by a Gysin sequence, as follows:

∗ ∗ i •^χ i+2 π i+2 σ i+1 •^χ i+3 · · · → H (M; R) −−−→ H (M; R) −→e H (Mf; R) −→ H (M; R) −−−→ H (M; R) → · · ·

•^χ In the homomorphism Hi(M; R) −−→ Hi+2(M; R), one takes the cup prod- uct with χ, the Euler class of the 2-disk bundle associated to the S1-ﬁbration

πe : Mf → M. This bundle is topologically equivalent to the complex line bundle associated to πe : Mf → M. We will denote this bundle by πe : Le → M.

Le is also induced by the immersion of M into CP N , from the complex line bundle associated to the Hopf ﬁbration, which is O(−1) ∈ P ic(CP N ). The Euler class

−1 of O(−1) is its ﬁrst Chern class, which is [ π ωFS], where ωFS is the K¨ahler class of the metric on CP N . Because the Euler class of Le is the pull-back of that of O(−1)

43 N −1 and the metric on M is the pull-back of that on CP , we have χ = [ π ω], where ω is the K¨ahlerform of M.

We let L : Hi(M; R) → Hi+2(M; R) denote the Lefschetz operator, i.e. [α] 7→

[ω] ^ [α] = [ω ∧ α]. The hard Lefschetz theorem states that for 0 ≤ k ≤ m − 1, L m−k : Hk(M; R) → H2m−k(M; R) is an isomorphism. In particular,

L : Hi(M; R) → Hi+2(M; R) is injective for 0 ≤ i ≤ m − 1 and surjective for m − 1 ≤ i ≤ n − 2. It is trivially surjective (zero) for i = n − 1 and i = n.

1 i •^χ i+2 Up to the factor − π , the homomorphism H (M; R) −−→ H (M; R) in the

Gysin sequence is the Lefschetz operator. By the hard Lefschetz theorem, it is therefore injective for 0 ≤ i ≤ m − 1. This injectivity implies that σ∗ : Hi+1(Mf; R) →

i ∗ i+1 i+1 H (M; R) has trivial image, and this implies that πe : H (M; R) → H (Mf; R) is surjective. For 0 ≤ i ≤ m − 1, using again the injectivity of the Lefschetz operator

•^χ in the homomorphism Hi−1(M; R) −−→ Hi+1(M; R), we then have the following short exact sequence:

∗ i−1 •^χ i+1 π i+1 0 → H (M; R) −−→ H (M; R) −→e H (Mf; R) → 0. (3.8)

This implies that βi+1(M; R) = βi−1(M; R) + βi+1(Mf; R). The same is true of

βi−1(M; R), i.e. it is equal to βi−3(M; R)+βi−1(Mf; R). For 0 ≤ k ≤ m, we therefore have:

βk(M; R) = βk(Mf; R) + βk−2(Mf; R) + ··· + βk−2l(Mf; R) + ··· (3.9)

44 Similarly, because L : Hi(M; R) → Hi+2(M; R) is surjective for m − 1 ≤ i ≤

∗ i+2 i+2 ∗ i+2 n−2, πe : H (M; R) → H (Mf; R) is zero. This implies that σ : H (Mf; R) →

Hi+1(M; R) is injective, and we have the following short exact sequence:

∗ i+2 σ i+1 •^χ i+3 0 → H (Mf; R) −→ H (M; R) −−→ H (M; R) → 0. (3.10)

For m ≤ k ≤ n this gives us βk(M; R) = βk+1(Mf; R) + βk+2(M; R), and like

(3.9), for m ≤ k ≤ n we then have:

βk(M; R) = βk+1(Mf; R) + βk+3(Mf; R) + ··· + βk+2l+1(Mf; R) + ··· (3.11)

This leads to the following proposition, which is the basic inequality between the total curvature of a complex projective manifold M and its Betti numbers. Part

A of Theorem 1.1 and the other inequalities in this chapter are based on this result:

Proposition 3.6. Let M be a compact complex manifold, of complex dimension m

(real dimension n = 2m) holomorphically immersed in the complex projective space

N CP , and let βi(M; R) be the Betti numbers of M with real coeﬃcients.

Then βm−1(M; R) + 2βm(M; R) + βm+1(M; R) ≤ T (M).

1 Proof. Let πe : Mf → M be the S -bundle over M induced the the Hopf ﬁbration as above. By Theorem 2.4.A and Propositions 3.1 and 3.5, we have:

45 n+1 X βm−1(M; R) + 2βm(M; R) + βm+1(M; R) = βj(Mf; R) ≤ T (Mf) = T (M). j=0 (3.12)

Remark 3.7. The calculations for (3.12) are slightly diﬀerent, depending on whether

M has even or odd complex dimension:

m m P2 P2 If m is even, then βm(M; R) = β2j(Mf; R) = β(n + 1 − 2j)(Mf; R). Because j=0 j=0

β0(Mf; R) and βn+1(Mf; R) are 1, this implies that βm(M; R) ≥ 1. If m is odd, m−1 m−1 P2 P2 then βm−1(M; R) = β2j(Mf; R) and βm+1(M; R) = β(n + 1 − 2j)(Mf; R). Be- j=0 j=0 cause β0(Mf; R) = 1, we have βm−1(M; R) ≥ 1, and because βn+1(Mf; R) = 1 we have βm+1(M; R) ≥ 1. The powers of the K¨ahlerform give M a non-trivial cohomology class in each of its even-dimensional cohomology groups and imply that its even-dimensional Betti numbers are non-zero - these observations can be seen as a reﬂection of this fact.

By the hard Lefschetz theorem, βm−1(M; R) = βm+1(M; R), so we can also state the conclusion of Proposition 3.6 as:

T (M) β (M; ) + β (M; ) ≤ . (3.13) m R m±1 R 2

46 The middle-dimensional Betti number of M, and the Betti numbers in the dimensions m ± 1, are the largest in dimensions with their respective parities. This gives us the following family of results:

Proposition 3.8. Let M be as in Proposition 3.6. Then the sum of any even and

T (M) any odd-dimensional real Betti number of M is bounded above by 2 .

T (M) If β2k(M; R) + β2l+1(M; R) = 2 , then all even-dimensional real Betti numbers of M in dimensions 2k through n − 2k are equal to β2k(M; R), and all odd- dimensional real Betti numbers of M in dimesions 2l + 1 through n − 2l − 1 are equal to β2l+1(M; R).

In particular, all of the even-dimensional real Betti numbers of M are bounded

T (M) above by 2 , and all of the odd-dimensional real Betti numbers of M are bounded

T (M) above by 2 −1. If equality holds for an even-dimensional Betti number β2k(M; R), then all odd-dimensional real Betti numbers of M are equal to 0, and if equality holds for an odd-dimensional Betti number β2l+1(M; R), then all even-dimensional real Betti numbers of M are equal to 1.

Proof. The observations above immediately imply that the sum of an even and an

T (M) odd Betti number of M is bounded above by 2 . The statement that β2k(M; R)+

T (M) β2l+1(M; R) = 2 is equivalent to the statement that β2k(M; R) + β2l+1(M; R) +

1 βn−2k(M; R)+βn−2l−1(M; R) = T (M). Letting Mf be the S -bundle over M as above,

Proposition 3.5 gives us the following relationship between the total curvature and

47 Betti numbers of Mf:

k l k l X X X X β2j(Mf; R)+ β2j+1(Mf; R)+ βn+1−2j(Mf; R)+ βn−2j(Mf; R) = TS2N+1 (Mf). j=0 j=0 j=0 j=0 (3.14)

Part A of Theorem 2.4 implies that the Betti numbers of Mf other than those in

(3.14) are zero. Proposition 3.5 then implies that Betti numbers of M in the ranges described above are constant.

As a corollary of these results, we have the following statement, which gives a parallel to the ﬁrst Chern-Lashof theorem for complex projective manifolds:

Theorem 1.1.A Let M be a compact complex manifold, of complex dimension m, holomorphically immersed in complex projective space. Let T (M) be its total absolute curvature and βi its Betti numbers with real coeﬃcients.

2m P m+1 Then βi ≤ ( 2 )T (M). In particular, T (M) ≥ 2. i=0

T (M) Proof. Each of the terms β2i−1 + β2i, for i = 1, 2, ··· , m, is bounded above by 2 .

β0 = 1 is likewise, and this implies the result.

2m P m+1 The statement that βi≤( 2 )T (M) in Theorem 1.1.A is sharp, in that equality i=0 holds for linearly embedded complex projective subspaces, but these are the only complex projective manifolds for which this equality holds. This follows from Part B of Theorem 1.1. It would be more interesting to characterize equality in Proposition

48 3.6. In their second paper, Chern and Lashof gave the following characterization of equality in their ﬁrst theorem:

Theorem 3.9 (Chern-Lashof, [CL58]). Let M n be a compact manifold immersed

n P in Euclidean space with βi(M; R) = T (M). Then the integral homology groups of i=0 M are torsion-free.

By Proposition 2.8, the equivalent theorem holds for submanifolds of spheres.

Equality in Proposition 3.6 therefore implies that the integral homology groups of the spherical pre-image Mf are torsion-free.

We note that if M is a complex projective manifold which is a complete intersection, Proposition 3.6 implies an inequality between the total curvature of M and its degree, because one can compute the Betti numbers of M in terms of the degrees of equations deﬁning its ideal.

We also note that the proofs of the ﬁrst Chern-Lashof theorem and the corresponding result for submanifolds of spheres in Theorem 2.4.A actually imply the following stronger result: Letting c(M) denote the minimum number of cells in a cell complex homotopy equivalent to M, c(M) ≤ T (M). Chern and Lashof discuss this in their second paper [CL58]. Spheres and complex projective spaces can be immersed with the minimum possible total absolute curvature, equal to 2, in The- orems 2.4 and 1.1 - we will discuss this in Chapter 4. c(Sn) = 2 for all n, however

49 c(CP m) = m + 1. This implies that the stronger conclusion above does not extend directly to complex projective manifolds.

50 Chapter 4

Complex Projective Manifolds with Minimal Total Curvature

The complex projective manifolds with minimal total absolute curvature are char- acterized as follows:

Theorem 1.1.B Let M be a compact complex manifold, holomorphically immersed in complex projective space. If T (M) < 4, then in fact T (M) = 2. This occurs precisely if M is a linearly embedded complex projective subspace.

In Example 5.7, we will see that this result is sharp.

1 Proof of Part B of Theorem 1.1. Let πe : Mf → M be the S -bundle over M which is induced from the Hopf ﬁbration, as in Chapter 3. By Proposition 3.1, Mf is

2N+1 immersed in S with TS2N+1 (Mf) < 4. Mf has odd dimension 2m + 1, so by

Theorem 2.7, Mf is homeomorphic to S2m+1. The Gysin sequence for the ﬁbration

51 2m+1 πe : S → M with integral cohomology is therefore as follows:

σ∗ •^χ π∗ · · · → Hi+1(S2m+1; Z) −→ Hi(M; Z) −−→ Hi+2(M; Z) −→e Hi+2(S2m+1; Z) → · · ·

As in the proof of Proposition 3.5, χ is the Euler class of the 2-disk bundle associated to the S1-ﬁbration Mf → M. Because Hi(S2m+1; Z) = 0 for i 6= 0, 2m + 1, we have that • ^ χ : Hi(M; Z) → Hi+2(M; Z) is an isomorphism for i = 0, 1, 2,

··· , 2m − 2. This implies that for k = 0, 1, 2, ··· , m, H2k(M; Z) is inﬁnite cyclic, generated by χk.

1 In the proof of Proposition 3.5, we observed that χ is equal to [− π ω], where ω

1 m is the K¨ahlerform of M. This implies that [− π ω] generates the top-dimensional

2m N 1 m cohomology H (M; Z) of M. If M is a degree d subvariety of CP , then [− π ω]

2m 1 m 2m N is d times a generator in H (M; Z), because [ π ωFS] generates H (CP ; Z). We

1 m 2m have seen that [− π ω] generates H (M; Z), so d = 1 and M is a linearly embedded subspace of CP N .

If CP m is a linearly embedded subspace of CP N , its pre-image via the Hopf

2m+1 2N+1 m ﬁbration is a totally geodesic S in S . By Proposition 3.1, TCP N (CP ) =

2m+1 2m+1 TS2N+1 (S ), and by Part C of Theorem 2.4, TS2N+1 (S ) = 2.

Remark 4.1. The proof of Theorem 1.1.B implies that linear subspaces are the only

52 complex projective manifolds M whose pre-images Mf via the Hopf ﬁbration are integer homology spheres.

We note that the spherical pre-images Mf in the proofs of Parts A and B of

Theorem 1.1 are minimal submanifolds of S2N+1. In light of the results above and in Theorem 2.4.C, Mf is a totally geodesic submanifold of S2N+1 precisely if M is a linear subspace of CP N , in which case T (Mf) = 2. Otherwise, Mf has total absolute curvature at least 4. In [Si68], Simons proved that all n-dimensional closed minimal subvarieties of the round sphere SN have index at least N − n, and nullity at least

(n + 1)(N − n), with equality only for totally geodesic round subspheres. It would be interesting to know if the apparent similarity between these statements is an indication that there are other results in this direction.

We also note that for many m ≥ 2, it is known that complex projective space is not the only m-dimensional compact complex manifold whose real cohomology ring is isomorphic to R[α]/[α]m+1, with [α] a cohomology class in H2(M; R). Compact complex manifolds which have the same real Betti numbers as CP m are known as fake projective spaces. In complex dimension 2, there are known to be ﬁfty fake projective planes, up to homeomorphism, and one hundred up to biholomorphism.

These were classiﬁed by work of Prasad and Yeung and Cartwright and Steger in

[PY07] and [CS10], and all of them can be realized as smooth algebraic surfaces.

Part B of Theorem 1.1 implies that the total absolute curvature of these spaces,

53 when realized as complex projective manifolds, is at least twice that of complex projective space.

We end this chapter by proving Proposition 1.2, which shows that the total absolute curvature has the same meaning for a complex projective manifold as

Chern and Lashof’s invariant for submanifolds of Euclidean space:

Proposition 1.2 Let M be a compact complex manifold holomorphically im-

N ⊥ < π N mersed in CP , and let Exp : ν 2 M → CP denote the normal exponential map. Then: Z Z 2 ⊥ 2 ⊥ −1 < π N T (M) = |det(dExp )|dV ol 2 = ](Exp ) (q)dV ol P . V ol(CP N ) ν M V ol(CP N ) C

< π N ν 2 M CP

Proof of Proposition 1.2. We begin by verifying the ﬁrst equation, that: Z 2 ⊥ < π T (M) = |det(dExp )|dV ol 2 . V ol(CP N ) ν M

< π ν 2 M

Let ~u be a unit normal vector to M at p, and let e1, ..., en be an orthonormal basis of principal vectors of the second fundamental form A~u with principal curvatures κ1, ..., κn. Let u2 = J(~u), u3, ..., u2N − n be an orthonormal basis for the

N orthogonal subspace to ~u in νpM, with J the complex structure of CP .

N Letting γ~u be the geodesic of CP through ~u, and E1, ..., En the parallel vector

ﬁelds along γ~u with initial conditions e1, ..., en, and F2, ..., F2N−n the parallel vector

54 ﬁelds along γ~u with initial conditions u2, u3, ..., u2N−n, we have:

⊥ • (dExp )r~u(ei) = ( cos(r) − κi sin(r))Ei(r) for i = 1, ..., n.

⊥ sin(2r) sin(r) cos(r) • (dExp )r~u(u2) = ( 2r )F2(r) = ( r )F2(r).

⊥ sin r • (dExp )r~u(uj) = ( r )Fj(r) for j = 3, ..., 2N − n.

⊥ 0 • (dExp )r~u(~u) = γ~u(r).

We therefore have:

n ⊥ Q cos(r) sin(2N−n−1)(r) det(dExp )r~u = (cos(r) − κi sin(r))( r2N−n−1 ) i=1

cos(r) sin(2N−n−1)(r) = det cos(r)IdTpM − sin(r)A~u ( r2N−n−1 ).

π For r less than 2 , this implies that:

⊥ sin(2N−n−1)(r) cos(r) |det(dExp )| = |det cos(r)IdTpM − sin(r)A~u |( r2N−n−1 ).

This implies the ﬁrst equation in Proposition 1.2: Z 2 ⊥ < π T (M) = |det(dExp )|dV ol 2 . V ol(CP N ) ν M

< π ν 2 M The veriﬁcation of the second equation is similar to the veriﬁcation of the corresponding fact for submanifolds of spheres, in the proof of Parts A and B of Theorem

2.4:

55 We let νMe → M be the bundle over M whose ﬁbre at p is the linear sub-

N−m N space CP of CP orthogonal to M at p. (We can also view νMe as the bundle whose ﬁbres are the cones over the projectivized normal spaces to M.) As

⊥ < π N in the proof of Theorem 2.4.A and B, Exp : ν 2 M → CP extends to a map

⊥ N ⊥ Expg : νMe → CP , and because the regular values of Exp contain those of ⊥ Expg , in addition to being of full measure, they contain an open, dense subset of

N N ⊥ < π N CP . We let CPreg denote the regular values of Exp : ν 2 M → CP .

As in the proof of Theorem 2.4.A and B, if ~v is a regular point of Exp⊥, there are neighborhoods V of ~v and Q of Exp⊥(~v) such that Exp⊥ : V → Q is a dif- feomorphism. As in the proof of Theorem 2.4.A and B, this allows us to deﬁne

< π a positive measure dµ on ν 2 M which is absolutely continuous with respect to

N ⊥ dV ol < π and is the pull-back of the measure on P via Exp , by integrating a ν 2 M C

⊥ measurable function φ against |det(dExp )|dV ol < π . For neighborhoods V and ν 2 M

Q with Exp⊥ : V → Q a diﬀeomorphism as above, we have:

Z Z Z ⊥ ⊥ −1 φdµ = φ|det(dExp )|dV ol < π = φ ◦ (Exp ) dV ol N . ν 2 M CP

V V Q

⊥ −1 N This implies that (Exp ) (CPreg) is of full measure with respect to dµ, and this implies that:

56 Z Z 2 ⊥ 2 ⊥ < π < π |det(dExp )|dV ol 2 = |det(dExp )|dV ol 2 V ol(CP N ) ν M V ol(CP N ) ν M

< π ⊥ −1 N ν 2 M (Exp ) (CPreg) Z 2 ⊥ −1 = ](Exp ) (q)dV ol P N V ol(CP N ) C

N CPreg N N Because CPreg is of full measure in CP , this implies the result:

Z Z 2 ⊥ 2 ⊥ −1 < π N |det(dExp )|dV ol 2 = ](Exp ) (q)dV ol P V ol(CP N ) ν M V ol(CP N ) C

< π N ν 2 M CP

Remark 4.2. Proposition 1.2 illustrates one of the diﬀerences between the characterization of minimal total absolute curvature for complex projective manifolds, in Part

B of Theorem 1.1, and the corresponding result for submanifolds of spheres in Part

C of Theorem 2.4: If Sn is a totally geodesic subsphere of SN , then almost all points

N ⊥ <π n N qe in the ambient space S have precisely two pre-images via Exp : ν S → S .

On the other hand, if CP m is a linear subspace of CP N , then almost all q in the

N ⊥ < π m N ambient CP have precisely one pre-image via Exp : ν 2 CP → CP . Thus,

⊥ <π n N N ⊥ < π m N N Exp : ν S → S covers S twice and Exp : ν 2 CP → CP covers CP once.

57 Chapter 5

Total Curvature and the

Geometry of Complex Projective

Hypersurfaces

We can give a more detailed explanation of the relationship between the total absolute curvature of a complex projective hypersurface and other aspects of its geometry. The key to these results is the following observation:

Proposition 5.1. Let M be a complex submanifold of a K¨ahlermanifold P , and let J denote its complex structure. Let ~u be a normal vector to M at a point p, and

∗ let e and e = J(e) be principal vectors for the second fundamental form A~u, with principal curvatures κ and −κ, as in Proposition 3.3.

58 ∗ ∗ Then e√+e and −e√+e are principal vectors for J(~u), with principal curvatures κ 2 2

θ θ ∗ and −κ respectively. In general, letting ~uθ = cos(θ)~u+sin(θ)J(~u),(cos( 2 )e+sin( 2 )e )

θ ∗ θ and (− sin( 2 )e + cos( 2 )e) are principal vectors for ~uθ with principal curvatures κ and −κ respectively.

Proof. Let U be a normal vector ﬁeld on a neighborhood of p which extends ~u. Be- cause P is K¨ahler,so that its connection ∇P commutes with the complex structure

J, we have:

P > P > ∗ AJ(~u)(e) = − ∇e J(U) = −J(∇e U) = −J( −A~u(e) ) = J(κe) = κe .

∗ ∗ Similarly, A (e∗) = −κ(−e) = κe. This implies that e√+e and −e√+e are J(~u) 2 2 principal vectors for AJ(~u):

∗ ∗ ∗ ∗ A ( e√+e ) = κ( e√+e ), A ( −e√+e ) = −κ( −e√+e ). J(~u) 2 2 J(~u) 2 2

More generally, we have:

θ θ ∗ θ ∗ θ ∗ A~uθ (cos( 2 )e + sin( 2 )e ) = cos( 2 )κ(cos(θ)e + sin(θ)e )+ sin( 2 )κ(sin(θ)e − cos(θ)e )

θ θ θ θ ∗ = κ(cos(θ) cos( 2 ) + sin(θ) sin( 2 ))e + κ(sin(θ) cos( 2 ) − sin( 2 ) cos(θ))e

θ θ ∗ θ θ ∗ = κ(cos(θ − 2 )e + sin(θ − 2 )e ) = κ(cos( 2 )e + sin( 2 )e ).

θ θ ∗ This implies that cos( 2 )e + sin( 2 )e is a principal vector for A~uθ with principal

θ ∗ θ curvature κ. By a similar calculation, or by Proposition 3.3, − sin( 2 )e + cos( 2 )e is a principal vector with principal curvature −κ.

59 For each complex line in the normal space of a K¨ahlersubmanifold, this gives us a set of tangent complex lines:

Deﬁnition 5.2 (Holomorphic Principal Directions). Let M be a complex submanifold of a K¨ahlermanifold P . Let ~u be a normal vector to M at p, and let e1, J(e1), e2, J(e2), ··· , em, J(em) be an orthonormal basis of principal vectors for A~u, with principal curvatures κ1, −κ1, κ2, −κ2, ··· , κm, −κm as in Proposition 3.3.

We will refer to the complex lines SpanC(ei) in the tangent space TpMas a set of holomorphic principal directions for the complex line SpanC(~u) in the normal space

νpM.

We let νP rojM denote the projectivized normal bundle of M in CP N , as in the proof of Proposition 1.2. Proposition 5.1 allows us to express the total absolute curvature of a complex projective manifold as an integral over νP rojM:

Proposition 5.3. Let M be a compact complex manifold of complex dimension m holomorphically immersed in CP N , and let νP rojM be its projectivized normal bundle. For each complex line in the normal space to M at a point p, as in Deﬁnition

2 2 2 5.2, let κ1, κ2, ··· , κm be the squares of the principal curvatures for a set of holo-

2 th 2 morphic principal directions, and let σi(κ ) be the i symmetric function of the κi , as in (3.6). Then:

60 π Z Z2 m 4π Y 2 2 2 (2N−2m−1) T (M) = N ( cos (r) − κ sin (r)) cos(r) sin (r) dr dV ol P roj V ol(CP ) | i | ν M i=1

νP roj M 0 π Z Z2 m 4π X i (2N−2m−1+2i) (2m−2i+1) 2 = N (−1) sin (r) cos (r)σi(κ ) dr dV ol P roj . V ol(CP ) | | ν M i=0

νP roj M 0 The factor 4π in Proposition 5.3, instead of 2 as in Deﬁnition 1.4, is because the

1 P roj ﬁbres of νp M over νp M have length 2π.

For a complex projective hypersurface M, this allows us to express the total absolute curvature as an integral over M itself:

Theorem 5.4. Let M be a compact complex manifold, of complex dimension m,

m+1 holomorphically immersed in CP . For each p in M, let K1, K2, ··· , Km be the holomorphic sectional curvatures of a family of holomorphic principal directions at

th p. Let σi(K) represent the i elementary symmetric function of the Ki.

2 Then Ki = 4 − 2κi , and the total absolute curvature of M is:

π Z Z2 m 2i 4π X 2 (m−i) sin (r) (2N−2m−1) T (M) = N | 1 − 3 sin (r) σi(K)| cos(r) sin (r) dr dV olM . V ol(CP ) 2i i=0

M 0

61 Proof. By the Gauss formula, the holomorphic sectional curvature Ki of a holomor-

2 phic principal direction SpanC(ei) is equal to 4−2κi , where κi, −κi are the principal curvatures of the principal vectors ei,J(ei). Substituting this in the expression for

T (M) in Proposition 5.3, we have:

π Z Z2 m 4π Y 2 4 − Ki 2 (2N−2m−1) T (M) = N | cos (r) − sin (r) | cos(r) sin (r)dr dV olM V ol(CP ) 2 i=1

M 0 π Z Z2 m 2 4π Y 2 sin (r) (2N−2m−1) = N | 1 − 3 sin (r) + Ki | cos(r) sin (r)dr dV olM V ol(CP ) 2 i=1

M 0 π Z Z2 m 2i 4π X 2 (m−i) sin (r) (2N−2m−1) = N | 1 − 3 sin (r) σi(K)| cos(r) sin (r)dr dV olM . V ol(CP ) 2i i=0

M 0

Calabi proved in [Ca53] that if a K¨ahlermanifold admits a holomorphic isometric immersion into a complex space form, even locally, then this immersion is essentially unique:

Theorem 5.5 (Calabi, [Ca53]). Let (M, h) be a K¨ahlermanifold of complex dimension m, and let U be a neighborhood in M which admits a holomorphic isometric immersion into a complex space form F (N, λ), of complex dimension N and holomorphic sectional curvature λ. Suppose that the image of this immersion does not

62 lie in any proper linear subspace of F (N, λ).

Then the dimension N of the ambient space is uniquely determined by the holomorphic sectional curvature λ and the metric h on U, and the immersion is uniquely determined up to a holomorphic isometry of F (N, λ).

Corollary 5.6. All holomorphic isometric immersions of a compact K¨ahlermanifold (with a ﬁxed metric) into complex projective space have the same total absolute curvature.

In general, Proposition 5.4 does not give a completely intrinsic representation of T (M) because it does not characterize the holomorphic principal directions of

M in CP N intrinsically. However it follows from Calabi’s theorem that the total absolute curvature of a complex projective manifold is actually part of its intrinsic geometry, and for curves Σ in CP 2, Proposition 5.4 does give a completely intrinsic representation of T (Σ):

Theorem 1.5 Let Σ be a smooth curve in CP 2, with K the sectional curvature of its projectively induced metric. Then:

Z 1 (K − 4)2 + 4 T (Σ) = dA . π 6 − K Σ Σ

63 Recall that the sectional curvature of Σ is bounded above by 4, away from the value 6 at which the integrand would be undeﬁned.

Proof of Theorem 1.5. By Proposition 5.4, the pointwise contribution to T (Σ) from a point of Σ with curvature K is: π Z2 K |1 + − 3 sin2(r) | cos(r) sin(r)dr. (5.1) 2 0

K 2 π K 2 K When r = 0, 1 + ( 2 − 3) sin (r) = 1. When r = 2 , 1 + ( 2 − 3) sin (r) = 2 − 2, which is less than or equal to zero for K ≤ 4, with equality precisely when K = 4.

K 2 2 2 In general, 1 + ( 2 − 3) sin (r) = 0 precisely when sin (r) = 6−K , so when r = √ √ √ 2 2 2 arcsin( 6−K ). For K ∈ (−∞, 4], 6−K takes values in (0, 1], so arcsin( 6−K ) has

π √ ( 2 ) a well-deﬁned value in (0, 2 ]. Let αK denote this value of arcsin 6−K . We then

π evaluate the integral in (5.1) over the intervals [0, αK ] and [αK , 2 ]. The resulting pointwise value for the total absolute curvature is:

π ZαK Z 2 K K 1 + − 3 sin2(r) cos(r) sin(r)dr − 1 + − 3 sin2(r) cos(r) sin(r)dr 2 2

0 αK 1 K2 − 8K + 20 1 (K − 4)2 + 4 = ( ) = ( ) . 2 24 − 4K 8 6 − K

Substituting this in the expression for T (Σ) in Theorem 5.4, together with the

2 π2 fact that V ol(CP ) = 2 , gives the result.

64 2 2 2 2 Example 5.7. The conic F described by z0 +z1 +z2 = 0 in CP is isometric to a round

2-sphere with curvature 2. As a degree 2 curve, Area(F) = 2 × Area(CP 1) = 2π.

(More generally, as explained in Remark 3.4, Wirtinger’s inequality implies that a degree d curve in CP 2 has area d × Area(CP 1) = dπ.) By Theorem 1.5, we then have: Z 1 (2 − 4)2 + 4 1 T (F) = dA = × 2π × 2 = 4. π 6 − 2 Σ π F

This shows that Part B of Theorem 1.1 is sharp.

More generally, for smooth degree d curves in CP 2, the following holds:

2 Proposition 5.8. Let Σd be a smooth curve of degree d in CP . Then:

2 2 2d − 4d + 4 ≤ T (Σd) ≤ 2d . (5.2)

(d−1)(d−2) Proof. Σd is a compact Riemann surface of genus 2 . As explained in Ex- ample 5.7, its area is dπ. Let Kd be the average sectional curvature of Σd. By the

Gauss-Bonnet formula,

dπKd = 2πχ(Σd) = 2π(2 − (d − 1)(d − 2)) = 2πd(3 − d).

Therefore, Kd = 2(3 − d).

65 (K−4)2+4 By applying Jensen’s inequality to the function 6−K in Theorem 1.5, which is convex on (−∞, 4], we then have: Z 2 2 (Kd − 4) + 4 (K − 4) + 4 Area(Σd) ≤ dAΣd = π × T (Σd). 6 − Kd 6 − K

Σd

2 This implies that 2d − 4d + 4 ≤ T (Σd).

(K−4)2+4 On the other hand, 6−K is asymptotic, as K → −∞, to a line with slope

−1. It is straight-forward to check that the linear function f(K) = 6 − K gives an

(K−4)2+4 upper bound for 6−K for all K ≤ 4, with equality when K = 4, and that this is the best linear upper bound possible, in that a line with greater slope or smaller

(K−4)2+4 intercept will no longer give an upper bound for 6−K for all K ∈ (−∞, 4]. We then have: Z Z 1 (K − 4)2 + 4 1 1 T (Σ ) = dA ≤ (6 − K)dA = (6Area(Σ ) − 2πχ(Σ )) = 2d2. d π 6 − K Σ π Σ π d d

Σd Σd

Hulin proved in [Hu00] that if a K¨ahler-Einsteinmetric on a compact complex manifold is induced by a holomorphic embedding into complex projective space, then the metric has positive scalar curvature. In particular, the only constant- curvature metrics on plane algebraic curves, with K ≡ Kd, occur on curves of degree 1, and on curves of degree 2 which are isometric (and thus congruent, by

Calabi’s theorem) to the curve in Example 5.7. Other than for these curves, the

66 ﬁrst inequality in Proposition 5.8 is strict. The second inequality in Proposition 5.8 is strict except for degree 1 curves.

A smooth degree d curve in CP 2 whose curvature is nearly constant will have total absolute curvature close to the lower bound 2d2 − 4d + 4 in Proposition 5.8. The equivalent statement for the upper bound 2d2 seems harder to formulate. For exam-

d d d ple, it is shown by Vitter in [Vi74] that the degree d Fermat curve z0 + z1 + z2 = 0, for d ≥ 3, has 3d points at which its sectional curvature is maximal, equal to 4.

This shows that for curves of arbitrarily high degree, one cannot replace the linear upper bound 6 − K in the proof of Proposition 5.8 by a stronger upper bound, even though for any > 0, a better linear upper bound is available for K conﬁned to the interval (−∞, 4 − ].

We now have two results which give a lower bound for the total absolute curvature of a smooth curve in CP 2 in terms of its degree - Proposition 3.6 and Propo- sition 5.8:

2 Let Σd be a smooth degree d curve in CP .

2 • By Proposition 3.6, 2d − 6d + 6 ≤ T (Σd).

2 2 • By Proposition 5.8, 2d − 4d + 4 ≤ T (Σd) ≤ 2d .

When d = 1, both results give the same estimate, that T (CP 1) ≥ 2. We know

67 1 that in fact T (CP ) = 2. When d = 2, Proposition 5.8 implies that T (Σ2) ≥ 4, which also follows from Part B of Theorem 1.1. Proposition 3.6 gives a weaker result, that T (Σ2) ≥ 2. Proposition 5.8 also implies that T (Σ2) < 8. When d = 3, Proposi- tion 3.6 implies that that T (Σ3) ≥ 6. Proposition 5.8 implies that 10 < T (Σ3) < 18.

For all d ≥ 3, the lower bound in Proposition 5.8 is stronger than the result in

Proposition 3.6. However, the upper and lower bounds in Proposition 5.8 together show that Proposition 3.6, based ultimately on the original Chern-Lashof theorems, gives the right order of growth in the degree d for the optimal lower bound for T (Σd).

Proposition 5.8 also shows that for all d ≥ 3, the total absolute curvatures of degree d curves are contained in an interval of length 4(d−1), between 2d2 −4d+4 and 2d2.

We note that the lower bound for degree d + 1 curves, 2(d + 1)2 − 4(d + 1) + 4, exceeds the upper bound for degree d curves, 2d2, by 2. This implies that the total absolute curvature of a smooth curve in CP 2 determines its degree. We record this in the following:

Proposition 1.6 Let Σ be a smooth curve in CP 2. Then the degree of Σ is the unique d ∈ N such that 2d2 − 4d + 4 ≤ T (Σ) ≤ 2d2.

By the Gauss-Bonnet formula, the (non-absolute) total curvature of a smooth plane curve determines its topology, and hence its degree for curves of degrees 3 and greater. However, the Gauss-Bonnet integral is the same for curves of degrees

68 1 and 2. Moreover, the Gauss-Bonnet integral is the same for all smooth curves of a ﬁxed topological type - it cannot distinguish between non-isomorphic curves with the same topology, or between geometrically distinct embeddings of isomorphic curves. Total absolute curvature can distinguish these things in some cases - for example, smooth conics which are not congruent to the curve in Example 5.7 will have total absolute curvature greater than 4. In this sense, total absolute curvature is a somewhat stronger invariant than total curvature.

Proposition 1.6 implies that the total absolute curvatures of complex subman-

2 ifolds of CP belong to disjoint intervals in R≥2, with each interval associated to curves of a ﬁxed degree. Smooth plane curves of degree d, together with their embeddings, belong to a continuous and connected family (parametrized by an open,

d+2−1 dense and connected subset of CP 2 ) so the total absolute curvatures of smooth

2 degree d curves in CP form a connected interval in R≥2. By Theorem 5.8 and Ex- ample 5.7, this interval is a point for d = 1, is a sub-interval of [4, 8) which includes the endpoint 4 for d = 2, and is a sub-interval of (2d2 − 4d + 4, 2d2) for d ≥ 3. De- termining these intervals precisely, describing total absolute curvature as a function on the parameter space of smooth degree d curves and studying the total absolute curvature of singular curves would strengthen these results.

One can ﬁx a higher-dimensional complex projective space and seek a charac-

69 terization of the total absolute curvatures of its closed complex submanifolds, along the lines of the results we have sketched above for complex submanifolds of CP 2.

However, one can also approach this question from a diﬀerent point of view: one can

ﬁx a compact complex manifold M, of projective type, and ask for the total absolute curvatures of all metrics on M which can be induced by holomorphic immersions into complex projective space.

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